Concept
23 (twenty-three) is the natural number following 22 and preceding 24. Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the fifth factorial prime, the second Woodall prime, and a happy number in decimal. It is an Eisenstein prime with no imaginary part and real part of the form It is also the fifth Sophie Germain prime and the fourth safe prime, and the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to for the integers 95, 119, 143, and 529. The third decimal repunit prime after R2 and R19 is R23, followed by R1031. 23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime. It is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down. The sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers. In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713). 23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem. 23 is the number of trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees. The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae. 23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.